15 Equation 906

In this chapter we study finite magmas that obey equation 906,

x = y ⋄ ((y ⋄ x) ⋄ (x ⋄ x))

for all $x, y$ . We can write this as

L_{y} (L_{y} x ⋄ S x) = x .

This implies that $L_{y}$ is surjective, hence invertible by finiteness, so

L_{y} x ⋄ S x = L_{y}^{- 1} x .

Corollary 15.1 Edge disjointness of left cycles

For any integer $n$ ,

L_{y} x = L_{z} x ⟹ L_{y}^{n} x = L_{z}^{n} x .

Proof ▶

This is trivial for $n = 0, 1$ , and $n = - 1$ follows from Equation 3. Observe that if the claim holds for $n = 0, - 1, \dots, - m$ for any $m \geq 1$ then it also holds for $n = - m - 1$ . Finally, since there is a common period to all the $L_{y}$ by finiteness (or Legendre’s theorem), the set of $n$ for which the claim holds is periodic. The claim follows.

Setting $n = N - 2$ for $N > 2$ a common period of $L_{y}, L_{z}$ (which exists by finiteness) we conclude that

L_{y} x = L_{z} x ⟹ L_{y}^{2} x = L_{z}^{2} x .

Theorem 15.2

✓

For finite magmas, equation 906 implies equation 3862,

(x ⋄ (x ⋄ x)) ⋄ x = x ⋄ x .

Proof ▶

Observe from Equation 2 that

L_{x} S^{2} x = L_{x} (L_{x} x ⋄ S x) = x

while from Equation 3 we have

L_{L_{x} S x} S^{2} x = L_{x} S x ⋄ S S x = L_{x}^{- 1} S x = x .

Thus

L_{x} S^{2} x = L_{L_{x} S x} S^{2} x = x

and hence by the $n = 2$ case of Corollary 15.1

L_{x} x = L_{L_{x} S x} x

giving the claim. □